Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

Critical Properties ofS=1 XXZ Chain

Kenn Kubo, Satoshi Takada

Published Feb 15, 1986• 35 citations
Publisher

Abstract

The S x - S x correlation function of S =1 XXZ chain is calculated at low temperatures by means of the transfer matrix with use of the Trotter formula. Critical properties of the correlation function, such as critical indices γ, η and ν and the mass identified by the inverse correlation length at T =0, are obtained. We conclude that a groundstate with an excitation gap appears for the anisotropy parameter Δ<Δ 1 ≃ -0.3 including the isotropic antiferromagnetic point (Δ=-1). We have obtained η≃1/4 at Δ=Δ 1 in accord with the theoretical prediction by Haldane and den Nijs. Our result, however, shows the monotonic increase of the excitation gap with decreasing Δ for Δ<Δ 1 contrary to Haldane's prediction.

DOI
10.1143/jpsj.55.438

Related papers

Critical Properties ofS=1/2 XXZ Chain

Satoshi Takada, Kenn Kubo

The S x S x correlation function of S =1/2 XXZ chain is numericaly obtained at low temperatures by means of the transfer matrix with use of the Trotter approximation. The inplane susceptibility and the correlation length are extrapolated to N =∞ where N is the number of the Trotter slicings. It is shown that the extrapolation works quite effectively and makes the estimation of the critical indices ν, γ and η possible for the anisotropy parameter -0.75≤Δ≤0.5. The obtained values of ν, γ and η satisfy the scaling relation and agree quite well with the analytic expression derived by Luther and Peschel.

Journal of the Physical Society of Japan•May 15, 1986•32 citations

The Spin Correlations of theS=3/2XXZChain

Kenn Kubo, Kazuo Saitoh, Satoshi Takada

Using the quantum transfer-matrix method we have calculated the spin correlation functions of the S =3/2 XXZ chain and examined the temperature dependence of the mass (≡ the inverse of the correlation length), susceptibility χ x ( Q ) and the magnetic structure factor S z ( Q ). The result shows that the mass vanishes at T =0 for the planar and isotropic cases (-1≦ Δ ≦1) in agreement with the theoretical prediction. At a low temperature the peak of S z ( Q ) at Q =π has been observed to persist even in the ferromagnetic region (0< Δ <0.8). The susceptibility χ z ( Q =0) at T =0 has been obtained with use of the extrapolation. The critical indices have been also examined.

Journal of the Physical Society of Japan•May 15, 1988•1 citation

Spin correlations in theS=1XXZchain

Kenn Kubo

The spin correlation functions as well as the string correlation functions introduced by den Nijs and Rommelse in the S=1 XXZ chain have been calculated by using the quantum transfer-matrix method. The temperatuare variation of the correlation lengths and the susceptibilities has been examined and found to differ according to the phase to which the ground state belongs. The excitation gap is estimated from the temperature dependence of these quantities.

Physical review. B, Condensed matter•Jul 1, 1992•27 citations

Thermal Properties of Chains of Antiferromagnetically Coupled Spins with s = 1: Numerical Evidence of the Haldane Gap at Nonzero Temperatures

Thomas Delica, Klaas Kopinga, Hajo Leschke +1 more

By the numerical transfer-matrix method we have computed several thermal properties for a Heisenberg chain of 150 antiferromagnetically coupled spins with quantum number s = 1. We dispensed with any extrapolation to the infinite chain, because the results for the magnetization, the susceptibility, the correlation length, and the static structure factor at wave number q = π have been found to be insensitive to a further increase of the chain length in the temperature regime considered. The characteristic features of our results, that occur at sufficiently low temperatures and low applied fields, can convincingly be explained by the existence of a gap in the excitation spectrum of the order of 0.4 coupling constants and, therefore, nicely complement previous numerical results for corresponding ground-state properties. We have also performed computations for a chain with an additional single-site anisotropy. We find, for appropriate parameter values, excellent agreement with experimental data for the quasi-one-dimensional model compound NENP.

Europhysics Letters (EPL)•May 1, 1991•43 citations

Scaling behavior in the S=1 antiferromagnetic X X Z chain (abstract)

J. C. Bonner, Gerhard H. Muller, John B. Parkinson

A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.

Journal of Applied Physics•Apr 15, 1988•0 citations

Critical behavior of the S=1 antiferromagnetic Heisenberg chain (abstract)

Jill C. Bonner

The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.

Journal of Applied Physics•Apr 15, 1991•1 citation
The S x - S x correlation function of S =1 XXZ chain is calculated at low temperatures by means of the transfer matrix with use of the Trotter formula. Critical properties of the correlation function, such as critical indices γ, η and ν and the mass identified by the inverse correlation length at T =0, are obtained. We conclude that a groundstate with an excitation gap appears for the anisotropy parameter Δ<Δ 1 ≃ -0.3 including the isotropic antiferromagnetic point (Δ=-1). We have obtained η≃1/4 at Δ=Δ 1 in accord with the theoretical prediction by Haldane and den Nijs. Our result, however, shows the monotonic increase of the excitation gap with decreasing Δ for Δ<Δ 1 contrary to Haldane's prediction.
The S x S x correlation function of S =1/2 XXZ chain is numericaly obtained at low temperatures by means of the transfer matrix with use of the Trotter approximation. The inplane susceptibility and the correlation length are extrapolated to N =∞ where N is the number of the Trotter slicings. It is shown that the extrapolation works quite effectively and makes the estimation of the critical indices ν, γ and η possible for the anisotropy parameter -0.75≤Δ≤0.5. The obtained values of ν, γ and η satisfy the scaling relation and agree quite well with the analytic expression derived by Luther and Peschel.
Using the quantum transfer-matrix method we have calculated the spin correlation functions of the S =3/2 XXZ chain and examined the temperature dependence of the mass (≡ the inverse of the correlation length), susceptibility χ x ( Q ) and the magnetic structure factor S z ( Q ). The result shows that the mass vanishes at T =0 for the planar and isotropic cases (-1≦ Δ ≦1) in agreement with the theoretical prediction. At a low temperature the peak of S z ( Q ) at Q =π has been observed to persist even in the ferromagnetic region (0< Δ <0.8). The susceptibility χ z ( Q =0) at T =0 has been obtained with use of the extrapolation. The critical indices have been also examined.
By the numerical transfer-matrix method we have computed several thermal properties for a Heisenberg chain of 150 antiferromagnetically coupled spins with quantum number s = 1. We dispensed with any extrapolation to the infinite chain, because the results for the magnetization, the susceptibility, the correlation length, and the static structure factor at wave number q = π have been found to be insensitive to a further increase of the chain length in the temperature regime considered. The characteristic features of our results, that occur at sufficiently low temperatures and low applied fields, can convincingly be explained by the existence of a gap in the excitation spectrum of the order of 0.4 coupling constants and, therefore, nicely complement previous numerical results for corresponding ground-state properties. We have also performed computations for a chain with an additional single-site anisotropy. We find, for appropriate parameter values, excellent agreement with experimental data for the quasi-one-dimensional model compound NENP.
A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.